So the homogeneous solution is
Now, we consider the non-homogeneous equation in differential operator form:
Since is the annihilator of , it follows that annihilates .
Therefore, when you apply the annihilator to both sides, we get
The characteristic equation is
The part in red generates the complimentary solution. What we're interested in is the term. This generates solutions with multiplicity 2 (but don't forget was a term in the complementary case!).
This implies that the particular solution will be of the form .
Now, substitute this into your original differential equation to solve for the constants A and B.
Once you get the particular solution, the general solution will be of the form .
It is at this point that you apply the initial conditions that you were given at the start of the problem.
Can you take it from here?